Numerically simulating structural behaviors of a product using explicit finite element analysis with a mass scaling enhanced subcycling technique

ABSTRACT

Methods and systems for numerically simulating structural behaviors of a product using explicit FEA with a mass scaling enhanced subcycling technique are disclosed. A FEA model of the product defined by a plurality of nodes and finite elements is received. A critical time step size is calculated for each finite element and then assigned to associated nodes. Elements are partitioned into N element groups with first group requiring minimum time step size Δt 1  and other element groups requiring respective time step sizes (Δt N =2 N−1 Δt 1 ). In order not to resort or repartition the finite elements and still obtain a stable solution, respective proper amounts of mass scaling are applied to those elements that have become too small to maintain a stable solution in their respective element groups. A time-marching simulation using explicit FEA with the mass scaling enhanced subcycling technique is then conducted with such a FEA model.

This application is related to a co-pending U.S. patent application Ser. No. 13/351,945 for “Numerically simulating structural behaviors of a product using explicit finite element analysis with a combined technique of subcycling and mass scaling” filed on Jan. 17, 2012, the content of which is incorporated herein by reference.

FIELD

The present invention generally relates to computer-aided engineering analysis, more particularly to methods and systems for numerically simulating structural behaviors of a product (e.g., automobile, airplane, etc.) using explicit finite element analysis (FEA) in conjunction with a mass scaling enhanced subcycling technique.

BACKGROUND

Finite element analysis (FEA) is a computerized method widely used in industry to model and solve engineering problems relating to complex systems such as three-dimensional non-linear structural design and analysis. FEA derives its name from the manner in which the geometry of the object under consideration is specified. With the advent of the modern digital computer, FEA has been implemented as FEA software. Basically, the FEA software is provided with a model of the geometric description and the associated material properties at each point within the model. In this model, the geometry of the system under analysis is represented by solids, shells and beams of various sizes, which are called elements. The vertices of the elements are referred to as nodes. The model is comprised of a finite number of elements, which are assigned a material name to associate the elements with the material properties. The model thus represents the physical space occupied by the object under analysis along with its immediate surroundings. The FEA software then refers to a table in which the properties (e.g., stress-strain constitutive equation, Young's modulus, Poisson's ratio, thermo-conductivity) of each material type are tabulated. Additionally, the conditions at the boundary of the object (i.e., loadings, physical constraints, etc.) are specified. In this fashion a model of the object and its environment is created.

FEA has two solution techniques: the implicit finite element analysis (“the implicit method”) and the explicit finite element analysis (“the explicit method”). Both methods are used to solve transient dynamic equations of motion and thus obtain an equilibrium solution to the equations. The methods march from time (t) through a discrete time interval or time interval Δt, to time (t+Δt). Such methods are sometimes referred to as time-marching simulation, which contains a number of consecutive time steps or solution cycles.

The present invention relates to the explicit method, which is stable only if time step size is very small—specifically, the time interval must be smaller than the time taken for an elastic wave to propagate from one side of an element to the other. The maximum time step for maintaining a stable solution in the explicit method is referred to as the critical time step size Δt_(cr). The speed of the elastic wave is a function of material mass and stiffness of the structure represented by the finite element and the element size or dimension. For a FEA model having substantially similar material, the smallest element generally controls the critical time step size.

Even one substantially smaller element in a FEA model can cause the critical time step size unnecessarily small for majority of the elements in the FEA model. This could happen when any element deforms to become too small in the middle of a time-marching simulation. As a result, a very small time step would be required for the remaining of the simulation. Not only would the simulation become very real time consuming, but also impractical. For example, it takes too long to obtain a simulated result for engineer to make design decision in time to meet the commercial need.

A number of prior art approaches to this problem have not been successful, for example, one approach referred to as subcycling that permits different time step sizes to be used in different parts of the finite element model. However, subcycling has its shortcomings and problems, for example, it requires periodically sorting of all elements of the entire FEA model into a number of groups being processed with dynamically changing time step sizes. This is not only difficult to implement in software, but also very costly. Periodical sorting of finite elements requires a significant amount of computer resource often defeating the originally purpose of speeding up the simulation.

Another approach is referred to as mass scaling, which increases the critical time step size of an element by artificially increasing its mass density. However, arbitrarily applying mass scaling to the FEA model can change the dynamic behaviors of the structure (i.e., artificially higher mass) in an unwanted manner.

It would therefore be desirable to have methods and systems for numerically simulating structural behaviors of a product using an explicit FEA with a combined technique of subcycling and mass scaling, such that the simulation is performed efficiently and effectively.

SUMMARY

This section is for the purpose of summarizing some aspects of the present invention and to briefly introduce some preferred embodiments. Simplifications or omissions in this section as well as in the abstract and the title herein may be made to avoid obscuring the purpose of the section. Such simplifications or omissions are not intended to limit the scope of the present invention.

The present invention discloses a system, method and software product for numerically simulating structural behaviors of a product using an explicit FEA with a mass scaling enhanced subcycling technique. According to an exemplary embodiment of the present invention, a finite element analysis (FEA) model is defined and received in a computer system having a finite element analysis application module installed thereon. The FEA model comprises a plurality of nodes and finite elements (e.g., shell, solid, foam elements, etc.). An element critical time step size (Δt_(cr)) is calculated for each finite element. Stable solution can be obtained in a time-marching engineering simulation using explicit FEA when the critical time step size is used. Critical time step size is calculated using Eq. 4 listed in the “DETAILED DESCRIPTION” section below.

Each node is also assigned a nodal critical time step size, which is the smallest Δt_(cr) of all finite elements it connects to. Then the finite elements are partitioned into N element groups based on nodal critical time step sizes. N is a whole number or integer greater than one with a first element group requiring a minimum time step size (Δt₁) and other element groups requiring respective time step sizes (Δt_(N)=2^(N−1)Δt₁). The required minimum time step size for each finite element is determined by the smallest nodal critical time step size. In other words, the required minimum time step size of a finite element is equal to the smallest nodal critical time step size of all nodes that define the finite element.

A time-marching simulation using explicit FEA in conjunction with a mass scaling enhanced subcycling technique is then conducted with such a FEA model. As a result, solution of the explicit FEA is obtained at major solution cycles and at sub solution cycles within each major solution cycle without periodical resorting of the finite elements. Mass scaling is applied to those elements that have become too small to maintain a stable solution in the respective element groups. In other words, the mass of those elements that become too small is numerically or artificially increased so that each of those elements can be processed with the peers in the originally assigned element group with the originally-calculated time step size. As a result, resorting is unnecessary during the simulation.

Objects, features, and advantages of the present invention will become apparent upon examining the following detailed description of an embodiment thereof, taken in conjunction with the attached drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

These and other features, aspects, and advantages of the present invention will be better understood with regard to the following description, appended claims, and accompanying drawings as follows:

FIG. 1 is a flowchart illustrating an exemplary process of numerically simulating structural behaviors of a product using explicit finite element analysis (FEA) in conjunction with a mass scaling enhanced subcycling technique, according to an embodiment of the present invention;

FIGS. 2A-2C are two-dimensional diagrams graphically showing an exemplary procedure for partitioning finite elements of a FEA model into different element groups in accordance with one embodiment of the present invention;

FIG. 3 is a diagram showing exemplary element groups and respective minimum time step sizes in accordance with one embodiment of the present invention;

FIG. 4 is a diagram showing two exemplary finite elements that can be used in FEA model representing a product (e.g., a car) in accordance with one embodiment of the present invention ;

FIG. 5 is a diagram showing an exemplary quadrilateral shell element and corresponding mass matrix and nodal acceleration vector used in explicit finite element analysis, to which mass scaling technique is applied in accordance with one embodiment of the present invention;

FIG. 6 is a diagram showing an exemplary time-marching simulation result of a car crash using explicit finite element analysis, according to one embodiment of the present invention; and

FIG. 7 is a function diagram showing salient components of an exemplary computer, in which one embodiment of the present invention may be implemented.

DETAILED DESCRIPTION

In the following description, numerous specific details are set forth in order to provide a thorough understanding of the present invention. However, it will become obvious to those skilled in the art that the present invention may be practiced without these specific details. The descriptions and representations herein are the common means used by those experienced or skilled in the art to most effectively convey the substance of their work to others skilled in the art. In other instances, well known methods, procedures, and components have not been described in detail to avoid unnecessarily obscuring aspects of the present invention.

Reference herein to “one embodiment” or “an embodiment” means that a particular feature, structure, or characteristic described in connection with the embodiment can be included in at least one embodiment of the invention. The appearances of the phrase “in one embodiment” in various places in the specification are not necessarily all referring to the same embodiment, nor are separate or alternative embodiments mutually exclusive of other embodiments. Further, the order of blocks in process flowcharts or diagrams representing one or more embodiments of the invention do not inherently indicate any particular order nor imply any limitations in the invention.

Embodiments of the present invention are discussed herein with reference to FIGS. 1-7. However, those skilled in the art will readily appreciate that the detailed description given herein with respect to these figures is for explanatory purposes as the invention extends beyond these limited embodiments.

FIG. 1 is a flowchart illustrating an exemplary process 100 of numerically simulating structural behaviors of a product using explicit finite element analysis (FEA) with a mass scaling enhanced subcycling technique, according to an embodiment of the present invention. Process 100 is preferably understood in conjunction with the previous figures and is implemented in software.

Process 100 starts by receiving a finite element analysis (FEA) model, which represents a product or structure (e.g., an automobile, an airplane, etc.) at step 102. The FEA model includes a plurality of nodes and a plurality of finite elements with associated material properties. Next, at step 104, element critical time step size is determined for each finite element (e.g., FIG. 2A). Nodes are assigned respective nodal critical time step sizes (e.g., FIG. 2B) accordingly. At a particular node, the smallest element critical time step size is selected for all finite elements it connects.

At step 106, finite elements of the FEA model are partitioned into N element groups based on nodal critical time step of nodes that define each finite element. N is a whole number or integer greater than one. In the N element groups, the first group of finite elements requires a minimum time step size Δt₁ either to satisfy the user's decision or to ensure a stable solution in explicit FEA (i.e., Δt_(cr)). Other element groups require minimum time step size set forth in the formula Δt_(n)=2^(n−1)Δt₁, where n=1, 2, . . . , N

In other words, the second group requires minimum time step size 2Δt₁, the third group requires 4Δt₁ and so forth, and the N-th group requires 2^(N−1)Δt₁. This ensures that nodal critical time steps of any finite element of the FEA model be integer multiple of one another.

Next at step 110, a time-marching simulation of the product represented by the FEA model is conducted using explicit FEA in conjunction with a mass scaling enhanced subcycling technique. The time-marching simulation contains a plurality of major solution cycles and a plurality of sub solution cycles within each major solution cycle. And each major solution cycle contains one or more sub solution cycles.

At the end of processing each element group, process 100 checks whether mass scaling is required for any finite element in that group. For those finite elements have deformed to become too small to maintain a stable solution in the originally assigned group, a proper amount of mass scaling is applied to ensure those finite elements can be processed with the originally assigned peers in that group at step 112. In other words, the mass density of any finite element deemed to be too small is increased artificially to ensure that element's critical time step size is at least equal to the original minimum time step size of the element group it belongs. For example, when an element's new critical time step size (Δt_(cr)) is smaller than the required minimum time step (Δt_(n)), the proper amount of mass density increase can be calculated using the difference of (Δt_(n)−Δt_(cr)) and Eq. 6 below.

This technique prevents periodical sorting/resorting the finite elements into different element groups thereby avoiding the problems or shortcomings of prior art approaches. Process 100 ends when a termination condition has reached at step 114, for example, the simulation time has exceeded a predefined total simulation time (e.g., user specified total simulation time). Otherwise process 100 moves back to step 110 to repeat until decision 114 becomes true.

FIGS. 2A-2C are two-dimensional diagrams graphically showing a series of steps for partitioning elements of an exemplary FEA model 200 into different element groups in accordance with one embodiment of the present invention. FEA model 200 contains various sizes of finite elements. First, element critical time step size of each finite element for obtaining a stable solution in explicit FEA is calculated using Eq. 4. Shown in FIG. 2A, smaller finite elements 202 that require a smaller critical time step size are denoted by “1”, while larger finite elements 204 are denoted by “2”. Then, nodal critical time step size of each node is determined by assigning the smallest element critical time step size of all elements it connects to. FIG. 2B shows nodal critical time step of the FEA model 200 as circles at respective nodes with “1”s 212 and “2”s 214. Attention is directed to finite elements denoted with “B” 210, which are finite elements around the boundary of two different sizes (i.e., two different critical time step sizes). The “B” or boundary finite elements 210 contain nodes having different nodal critical time step sizes. When applying subcycling technique, these boundary finite elements 210 need to be processed at the smaller time step. Element group “1” 222 and element group “2” 224 shown in FIG. 2C are the result of applying subcycling technique on the FEA model 200.

In a more general representation of partitioning finite elements into various different element groups, FIG. 3 shows a data structure used for element group management in accordance with one embodiment of the present invention. Partitioning of finite elements 303 results into N element groups 301, where N is a whole number or integer greater than one. Each element group 301 contains a list of finite elements 303. For example, the first group includes finite elements 310 a-e, the second group contains elements 320 a-b, the third group contains elements 330 a-d, and the N-th group contains elements 350 a-m. Minimum required time step sizes 302 for respective element groups 301 are integer multiples of one another. According to one embodiment, the first group requires minimum time step size Δt₁, the second element group requires 2Δt₁, the third element group requires 4Δt₁, and the N-th element group requires 2^(N−1)Δt₁. The a mass scaling enhanced subcycling technique keeps the list of finite elements within each of the N element groups the same throughout the entire time-marching simulation.

A number of different types of finite elements can be used for the FEA model, for example, 4-node quadrilateral shell element 401 and 8-node solid element 402 shown in FIG. 4.

Referring now to FIG. 5, the mass scaling technique is shown. In an explicit finite element analysis, the equation-of-motion of a system (e.g., a structure) is listed as follows: a=M ⁻¹(f _(ext) −f _(int))   (Eq. 1) where a is the unknown acceleration vector representing nodal accelerations, M is a lumped, diagonal, matrix with zeros in the off-diagonal terms, and f_(ext) and f_(int) are the external applied forces and the internal nodal forces, respectively. The force term f_(ext)−f_(int) is referred to as the residual forces including, but not limited to, applied loads, contact forces, internal stresses. Because M has only diagonal terms, computation of M⁻¹can simply be performed as a division instead of costly matrix inversion. FIG. 5 shows a diagonal mass matrix 510 and corresponding acceleration vector 520 for an exemplary 4-node shell element 500. In this exemplary mass matrix 510, each of the four nodes has the same amount of mass represented by the diagonal term m_(o) (e.g., lumped nodal mass 501, 502, 503, or 504). All of the off-diagonal terms 506 are zeros. The acceleration vector 520 includes the acceleration at each of the nodes (i.e., a₁ 521, a₂ 522, a₃ 523, and a₄ 524). The inertia force for each of the nodes is calculated with the diagonal terms multiplying the corresponding acceleration as follows:

$\begin{matrix} {{f_{i} = {\sum\limits_{{j = 1},N}{m_{ij}a_{j}}}}{{i = 1},N}} & \left( {{Eq}.\mspace{14mu} 2} \right) \end{matrix}$ where N is the number of nodes in the element. N is equal to 4 for a quadrilateral shell element and 8 for a hexahedral solid element. The inertia forces at four nodes of the element 500 are calculated in accordance with Eq. 2 as follows: f ₁ =m _(o) a ₁+(0)a ₂+(0)a ₃+(0)a ₄=m_(o) a ₁   (Eq. 3) f ₂=(0)a ₁ +m _(o) a ₂+(0)a ₃+(0)a ₄ =m _(o) a ₂ f ₃=(0)a ₁+(0)a ₂ +m _(o) a ₃+(0)a ₄ =m _(o) a ₃ f ₄=(0)a ₁+(0)a ₂+(0)a ₃ +m _(o) a ₄ =m _(o) a ₄

In order to simplify the illustration and description, only one component at each node is shown in FIG. 5. In a Cartesian coordinate system, there are three translational masses at each node, hence total number of components is 12 (i.e., three times four).

Eq. 1 is solved with an explicit method in time integration, which is performed with one time step at a time. The explicit method can only produce stable solution only if the time step size Δt is smaller than the critical value Δt_(cr) (i.e., critical time step size), which is determined by the following formula:

$\begin{matrix} {{\Delta\; t_{er}} = \frac{2}{\omega_{\max}}} & \left( {{Eq}.\mspace{14mu} 4} \right) \end{matrix}$

ω_(max) is the maximum eigenfrequency of the system. The eigenfrequencies can be obtained by solving the following eigenvalue problem: det[K−ω _(i) ² M]=0   (Eq. 5) where K and M are m×m stiffness and mass matrix of the system, respectively, and m is number of degrees of freedom of the system. The maximum eigenfrequency ω_(max) is generally a large number (e.g., in the order of 10⁶ Hz), thus the critical time step size Δt_(cr) is a very small number (e.g., in the order of microseconds). This small time step limits the usefulness of the explicit method in simulating a quasi-static process such as metal forming

One way to increase the critical time step size is to decrease the maximum eigenfrequency ω_(max), which can be achieved by increasing the mass of the system. The present invention allows a user to specify a desired time step size Δt . The required mass to ensure stability of the explicit method can then be calculated as follows: m=αk(Δt)²   (Eq. 6) where m is the required mass, k is the stiffness and α is a constant that depends on the specific integration scheme. In a finite element analysis model, each of the elements has a stiffness that depends on the geometry and material properties. Hence, the required mass may be computed using Eq. 6 for each of the elements at each solution cycle. The critical time step size is a function of the square root of the mass, thus a much larger mass is required to increase the critical time step. The required mass is also referred to as the added mass hereinafter, and mass scaling is referred to the procedure to modify the real mass of a system with the added mass.

Mass scaling can be applied with a number of techniques. One of the most basic approaches is to change material mass density of the entire structure in a trial-and-error manner. Such procedure can generally be performed by users of the FEA without any automated computation, and the FEA solution could be obtained with a suitable material mass density after several material mass density adjustments.

Automated procedure is implemented in an application module (e.g., FEA software). Generally, a desired time step size is specified by the user of the FEA, the application module comprises certain logics or subprograms that would calculate a factor to increase the mass density automatically such that the desired time step size can be used in the engineering simulation to obtain a solution. The automated procedure can be carried out in an element-by-element basis. For each element in a FEA model, a critical time step size can be calculated based on element's mass and geometry. Based on Eq. 4, the critical time step size Δt_(cr) can be increased by decreasing the maximum eigenfrequency, which is equivalent to increasing the mass of an otherwise unaltered element.

If a particular element's critical time step size is smaller than the desired time step size, an added mass can be calculated and added to the particular element to ensure the critical time step size of that element would become at least as large as the desired time step size. Some elements may not require any added mass when the critical time step size of those elements is larger or equal to the desired time step size, which can be specified by user (e.g., a parameter in the input file).

Referring now to FIG. 6, which is a diagram showing an exemplary time-marching simulation result of a car crash using explicit finite element analysis. In other words, a graphical display of numerical simulation results from a finite element analysis of a numerically simulated impact event. There are two structures included in the simulation: an automobile and a barrier. Both of the structures can be represented in a finite element analysis model. The automobile can be modeled with more than 1,000,000 finite elements (e.g., shell, solid and beam elements) with comparable number of nodes.

According to one aspect, the present invention is directed towards one or more computer systems capable of carrying out the functionality described herein. An example of a computer system 700 is shown in FIG. 7. The computer system 700 includes one or more processors, such as processor 704. The processor 704 is connected to a computer system internal communication bus 702. Various software embodiments are described in terms of this exemplary computer system. After reading this description, it will become apparent to a person skilled in the relevant art(s) how to implement the invention using other computer systems and/or computer architectures.

Computer system 700 also includes a main memory 708, preferably random access memory (RAM), and may also include a secondary memory 710. The secondary memory 710 may include, for example, one or more hard disk drives 712 and/or one or more removable storage drives 714, representing a floppy disk drive, a magnetic tape drive, an optical disk drive, etc. The removable storage drive 714 reads from and/or writes to a removable storage unit 718 in a well-known manner. Removable storage unit 718, represents a floppy disk, magnetic tape, optical disk, etc. which is read by and written to by removable storage drive 714. As will be appreciated, the removable storage unit 718 includes a computer usable storage medium having stored therein computer software and/or data.

In alternative embodiments, secondary memory 710 may include other similar means for allowing computer programs or other instructions to be loaded into computer system 700. Such means may include, for example, a removable storage unit 722 and an interface 720. Examples of such may include a program cartridge and cartridge interface (such as that found in video game devices), a removable memory chip (such as an Erasable Programmable Read-Only Memory (EPROM), Universal Serial Bus (USB) flash memory, or PROM) and associated socket, and other removable storage units 722 and interfaces 720 which allow software and data to be transferred from the removable storage unit 722 to computer system 700. In general, Computer system 700 is controlled and coordinated by operating system (OS) software, which performs tasks such as process scheduling, memory management, networking and I/O services.

There may also be a communications interface 724 connecting to the bus 702. Communications interface 724 allows software and data to be transferred between computer system 700 and external devices. Examples of communications interface 724 may include a modem, a network interface (such as an Ethernet card), a communications port, a Personal Computer Memory Card International Association (PCMCIA) slot and card, etc. The computer 700 communicates with other computing devices over a data network based on a special set of rules (i.e., a protocol). One of the common protocols is TCP/IP (Transmission Control Protocol/Internet Protocol) commonly used in the Internet. In general, the communication interface 724 manages the assembling of a data file into smaller packets that are transmitted over the data network or reassembles received packets into the original data file. In addition, the communication interface 724 handles the address part of each packet so that it gets to the right destination or intercepts packets destined for the computer 700. In this document, the terms “computer program medium” and “computer usable medium” are used to generally refer to media such as removable storage drive 714, and/or a hard disk installed in hard disk drive 712. These computer program products are means for providing software to computer system 700. The invention is directed to such computer program products.

The computer system 700 may also include an input/output (I/O) interface 730, which provides the computer system 700 to access monitor, keyboard, mouse, printer, scanner, plotter, and alike.

Computer programs (also called computer control logic) are stored as application modules 706 in main memory 708 and/or secondary memory 710. Computer programs may also be received via communications interface 724. Such computer programs, when executed, enable the computer system 700 to perform the features of the present invention as discussed herein. In particular, the computer programs, when executed, enable the processor 704 to perform features of the present invention. Accordingly, such computer programs represent controllers of the computer system 700.

In an embodiment where the invention is implemented using software, the software may be stored in a computer program product and loaded into computer system 700 using removable storage drive 714, hard drive 712, or communications interface 724. The application module 706, when executed by the processor 704, causes the processor 704 to perform the functions of the invention as described herein.

The main memory 708 may be loaded with one or more application modules 706 that can be executed by one or more processors 704 with or without a user input through the I/O interface 730 to achieve desired tasks. In operation, when at least one processor 704 executes one of the application modules 706, the results are computed and stored in the secondary memory 710 (i.e., hard disk drive 712). The status of the finite element analysis is reported to the user via the I/O interface 730 either in a text or in a graphical representation.

In one embodiment, an application module 706 is configured to facilitate the creation of a finite element analysis model representing a product. The application module 706 further allows the mass scaling enhanced subcycling technique to be used in conjunction with explicit finite element analysis. In another embodiment, the application module 706 is configured to facilitate mass scaling to certain deformed finite elements for maintaining stable solution with subcycling.

Although the present invention has been described with reference to specific embodiments thereof, these embodiments are merely illustrative, and not restrictive of, the present invention. Various modifications or changes to the specifically disclosed exemplary embodiments will be suggested to persons skilled in the art. For example, whereas quadrilateral shell finite elements have been shown and described as to how the mass scaling enhanced subcycling technique is accomplished, other types of finite element can be used instead, for example, 8-node hexahedral element, 4-node tetrahedral element, 3-node triangular element, etc. Further, whereas a simple FEA model 200 having two element groups has been shown and described to show the partitioning scheme, other FEA models having more than two element groups may be used for achieving the same. In summary, the scope of the invention should not be restricted to the specific exemplary embodiments disclosed herein, and all modifications that are readily suggested to those of ordinary skill in the art should be included within the spirit and purview of this application and scope of the appended claims. 

I claim:
 1. A method of numerically simulating structural behaviors of a product using explicit finite element analysis (FEA) with a mass scaling enhanced subcycling technique, said method comprising: receiving a FEA model in a computer system having an explicit FEA application module installed thereon, said FEA model representing a product defined by a plurality of nodes and a plurality of finite elements with associated properties; calculating an element critical time step size of each finite element; establishing a nodal critical time step for each node as the smallest one of respective element critical time step sizes of all connected finite elements to said each node; partitioning the plurality of finite elements into N element groups based on the smallest nodal critical time step size of said each finite element's nodes, where N is a whole number greater than one; conducting a time-marching simulation using the FEA model with the explicit FEA application module executed in the computer system, said explicit FEA application module applying a mass scaling enhanced subcycling technique for obtaining simulated structural behaviors at a number of major solution cycles and at one or more sub solution cycles within each major solution cycle, and said mass scaling enhanced subcycling technique applying mass scaling to ensure a stable solution within said each element group; and ending said time-marching simulation when a predetermined termination condition has reached.
 2. The method of claim 1, wherein the product comprises an automobile.
 3. The method of claim 2, wherein the time-marching simulation comprises a numerical simulation of crashworthiness of the automobile in an impact event.
 4. The method of claim 1, wherein said N element groups of finite elements are associated with respective minimum time step sizes.
 5. The method of claim 4, wherein each of said minimum time step sizes are calculated in accordance with formula Δt_(n)=2^(n−1)Δt₁, where n=1, 2, . . . , N, and Δt₁ is the smallest element critical time step size of the FEA model.
 6. The method of claim 4, wherein said applying mass scaling increases mass density of said finite elements such that said finite elements are processed with said respective minimum time step sizes.
 7. The method of claim 1, wherein the mass scaling enhanced subcycling technique further includes synchronizing the simulated structural behaviors obtained at end of said each major solution cycle for said finite elements requiring solutions at said sub solution cycles.
 8. The method of claim 1, wherein the predetermined termination condition comprises a user specified total simulation time.
 9. A non-transitory machine-readable computer readable storage medium containing instructions, when executed in a computer system, for numerically simulating structural behaviors of a product using explicit finite element analysis (FEA) with a mass scaling enhanced subcycling technique by a method comprising: receiving a FEA model in a computer system having an explicit FEA application module installed thereon, said FEA model representing a product defined by a plurality of nodes and a plurality of finite elements with associated properties; calculating an element critical time step size of each finite element; establishing a nodal critical time step for each node as the smallest one of respective element critical time step sizes of all connected finite elements to said each node; partitioning the plurality of finite elements into N element groups based on the smallest nodal critical time step size of said each finite element's nodes, where N is a whole number greater than one; conducting a time-marching simulation using the FEA model with the explicit FEA application module executed in the computer system, said explicit FEA application module applying a mass scaling enhanced subcycling technique for obtaining simulated structural behaviors at a number of major solution cycles and at one or more sub solution cycles within each major solution cycle, and said mass scaling enhanced subcycling technique applying mass scaling to ensure a stable solution within said each element group; and ending said time-marching simulation when a predetermined termination condition has reached.
 10. The non-transitory machine-readable computer readable storage medium of claim 9, wherein said N element groups of finite elements are associated with respective minimum time step sizes.
 11. The non-transitory machine-readable computer readable storage medium of claim 10, wherein each of said minimum time step sizes are calculated in accordance with formula Δt_(n)=2^(n−1)Δt₁, where n=1, 2, . . . , N, and Δt₁ is the smallest element critical time step size of the FEA model.
 12. The non-transitory machine-readable computer readable storage medium of claim 10, wherein said applying mass scaling increases mass density of said finite elements such that said finite elements are processed with said respective minimum time step sizes.
 13. The non-transitory machine-readable storage medium of claim 9, wherein the mass scaling enhanced subcycling technique further includes synchronizing the simulated structural behaviors obtained at end of said each major solution cycle for said finite elements requiring solutions at said sub solution cycles.
 14. A system for numerically simulating structural behaviors of a product using explicit finite element analysis (FEA) with a mass scaling enhanced subcycling technique, said system comprising: a main memory for storing computer readable code for an explicit finite element analysis (FEA) application module; at least one processor coupled to the main memory, said at least one processor executing the computer readable code in the main memory to cause the explicit FEA application module to perform operations by a method of: receiving a FEA model representing a product defined by a plurality of nodes and a plurality of finite elements with associated properties; calculating an element critical time step size of each finite element; establishing a nodal critical time step for each node as the smallest one of respective element critical time step sizes of all connected finite elements to said each node; partitioning the plurality of finite elements into N element groups based on the smallest nodal critical time step size of said each finite element's nodes, where N is a whole number greater than one; conducting a time-marching simulation using the FEA model with the explicit FEA application module executed in the system, said explicit FEA application module applying a mass scaling enhanced subcycling technique for obtaining simulated structural behaviors at a number of major solution cycles and at one or more sub solution cycles within each major solution cycle, and said mass scaling enhanced subcycling technique applying mass scaling to ensure a stable solution within said each element group; and ending said time-marching simulation when a predetermined termination condition has reached.
 15. The system of claim 14, wherein said N element groups of finite elements are associated with respective minimum time step sizes.
 16. The system of claim 15, wherein each of said minimum time step sizes are calculated in accordance with formula Δt_(n)=2^(n−1)Δt₁, where n=1, 2, . . . , N, and Δt₁ is the smallest element critical time step size of the FEA model.
 17. The system of claim 15, wherein said applying mass scaling increases mass density of said finite elements such that said finite elements are processed with said respective minimum time step sizes.
 18. The system of claim 14, wherein the mass scaling enhanced subcycling technique further includes synchronizing the simulated structural behaviors obtained at end of said each major solution cycle for said finite elements requiring solutions at said sub solution cycles. 